3.303 \(\int \frac {A+B \log (x)}{\sqrt {a+b \log (x)}} \, dx\)

Optimal. Leaf size=69 \[ \frac {\sqrt {\pi } e^{-\frac {a}{b}} (2 A b-B (2 a+b)) \text {erfi}\left (\frac {\sqrt {a+b \log (x)}}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {B x \sqrt {a+b \log (x)}}{b} \]

[Out]

1/2*(2*A*b-(2*a+b)*B)*erfi((a+b*ln(x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/exp(a/b)+B*x*(a+b*ln(x))^(1/2)/b

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Rubi [A]  time = 0.07, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2294, 2299, 2180, 2204} \[ \frac {\sqrt {\pi } e^{-\frac {a}{b}} (2 A b-B (2 a+b)) \text {Erfi}\left (\frac {\sqrt {a+b \log (x)}}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {B x \sqrt {a+b \log (x)}}{b} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[x])/Sqrt[a + b*Log[x]],x]

[Out]

((2*A*b - (2*a + b)*B)*Sqrt[Pi]*Erfi[Sqrt[a + b*Log[x]]/Sqrt[b]])/(2*b^(3/2)*E^(a/b)) + (B*x*Sqrt[a + b*Log[x]
])/b

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2294

Int[((A_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]*(B_.))/Sqrt[Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]*(b_.) + (
a_)], x_Symbol] :> Simp[(B*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/(b*e), x] + Dist[(2*A*b - B*(2*a + b*n))/
(2*b), Int[1/Sqrt[a + b*Log[c*(d + e*x)^n]], x], x] /; FreeQ[{a, b, c, d, e, A, B, n}, x]

Rule 2299

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[1/(n*c^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p
, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]

Rubi steps

\begin {align*} \int \frac {A+B \log (x)}{\sqrt {a+b \log (x)}} \, dx &=\frac {B x \sqrt {a+b \log (x)}}{b}+\frac {(2 A b-(2 a+b) B) \int \frac {1}{\sqrt {a+b \log (x)}} \, dx}{2 b}\\ &=\frac {B x \sqrt {a+b \log (x)}}{b}+\frac {(2 A b-(2 a+b) B) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\log (x)\right )}{2 b}\\ &=\frac {B x \sqrt {a+b \log (x)}}{b}+\frac {(2 A b-(2 a+b) B) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \log (x)}\right )}{b^2}\\ &=\frac {(2 A b-(2 a+b) B) e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \log (x)}}{\sqrt {b}}\right )}{2 b^{3/2}}+\frac {B x \sqrt {a+b \log (x)}}{b}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 80, normalized size = 1.16 \[ \frac {e^{-\frac {a}{b}} (2 A b-B (2 a+b)) \sqrt {-\frac {a+b \log (x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \log (x)}{b}\right )+2 B x (a+b \log (x))}{2 b \sqrt {a+b \log (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[x])/Sqrt[a + b*Log[x]],x]

[Out]

(2*B*x*(a + b*Log[x]) + ((2*A*b - (2*a + b)*B)*Gamma[1/2, -((a + b*Log[x])/b)]*Sqrt[-((a + b*Log[x])/b)])/E^(a
/b))/(2*b*Sqrt[a + b*Log[x]])

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(x))/(a+b*log(x))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [B]  time = 0.31, size = 129, normalized size = 1.87 \[ -\frac {\sqrt {\pi } A \operatorname {erf}\left (-\frac {\sqrt {b \log \relax (x) + a} \sqrt {-b}}{b}\right ) e^{\left (-\frac {a}{b}\right )}}{\sqrt {-b}} + \frac {\sqrt {\pi } B \operatorname {erf}\left (-\frac {\sqrt {b \log \relax (x) + a} \sqrt {-b}}{b}\right ) e^{\left (-\frac {a}{b}\right )}}{2 \, \sqrt {-b}} + \frac {\sqrt {\pi } B a \operatorname {erf}\left (-\frac {\sqrt {b \log \relax (x) + a} \sqrt {-b}}{b}\right ) e^{\left (-\frac {a}{b}\right )}}{\sqrt {-b} b} + \frac {\sqrt {b \log \relax (x) + a} B x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(x))/(a+b*log(x))^(1/2),x, algorithm="giac")

[Out]

-sqrt(pi)*A*erf(-sqrt(b*log(x) + a)*sqrt(-b)/b)*e^(-a/b)/sqrt(-b) + 1/2*sqrt(pi)*B*erf(-sqrt(b*log(x) + a)*sqr
t(-b)/b)*e^(-a/b)/sqrt(-b) + sqrt(pi)*B*a*erf(-sqrt(b*log(x) + a)*sqrt(-b)/b)*e^(-a/b)/(sqrt(-b)*b) + sqrt(b*l
og(x) + a)*B*x/b

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maple [F]  time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {B \ln \relax (x )+A}{\sqrt {b \ln \relax (x )+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(x))/(b*ln(x)+a)^(1/2),x)

[Out]

int((A+B*ln(x))/(b*ln(x)+a)^(1/2),x)

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maxima [B]  time = 0.74, size = 156, normalized size = 2.26 \[ \frac {\frac {2 \, \sqrt {\pi } A \operatorname {erf}\left (\sqrt {b \log \relax (x) + a} \sqrt {-\frac {1}{b}}\right ) e^{\left (-\frac {a}{b}\right )}}{\sqrt {-\frac {1}{b}}} - \frac {2 \, \sqrt {\pi } B a \operatorname {erf}\left (\sqrt {b \log \relax (x) + a} \sqrt {-\frac {1}{b}}\right ) e^{\left (-\frac {a}{b}\right )}}{b \sqrt {-\frac {1}{b}}} - \frac {{\left (\frac {\sqrt {\pi } b \operatorname {erf}\left (\sqrt {b \log \relax (x) + a} \sqrt {-\frac {1}{b}}\right ) e^{\left (-\frac {a}{b}\right )}}{\sqrt {-\frac {1}{b}}} - 2 \, \sqrt {b \log \relax (x) + a} b e^{\left (\frac {b \log \relax (x) + a}{b} - \frac {a}{b}\right )}\right )} B}{b}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(x))/(a+b*log(x))^(1/2),x, algorithm="maxima")

[Out]

1/2*(2*sqrt(pi)*A*erf(sqrt(b*log(x) + a)*sqrt(-1/b))*e^(-a/b)/sqrt(-1/b) - 2*sqrt(pi)*B*a*erf(sqrt(b*log(x) +
a)*sqrt(-1/b))*e^(-a/b)/(b*sqrt(-1/b)) - (sqrt(pi)*b*erf(sqrt(b*log(x) + a)*sqrt(-1/b))*e^(-a/b)/sqrt(-1/b) -
2*sqrt(b*log(x) + a)*b*e^((b*log(x) + a)/b - a/b))*B/b)/b

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {A+B\,\ln \relax (x)}{\sqrt {a+b\,\ln \relax (x)}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*log(x))/(a + b*log(x))^(1/2),x)

[Out]

int((A + B*log(x))/(a + b*log(x))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {A + B \log {\relax (x )}}{\sqrt {a + b \log {\relax (x )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(x))/(a+b*ln(x))**(1/2),x)

[Out]

Integral((A + B*log(x))/sqrt(a + b*log(x)), x)

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